1. Rút gọn:A= \(\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\)+\(\dfrac{\sqrt{x}-1}{\sqrt{x}+1}\)-\(\dfrac{3\sqrt{x}+1}{x-1}\)
2. B= \(\dfrac{\sqrt{x}}{\sqrt{x}+2}\)-\(\dfrac{2\sqrt{x}-4}{x-4}\) chứng minh B= \(\dfrac{\sqrt{x}-2}{\sqrt{x}+2}\)
3. C= \(\dfrac{x}{x-4}\)+\(\dfrac{1}{\sqrt{x}-2}\)+\(\dfrac{1}{\sqrt{x}+2}\) chứng minh C= \(\dfrac{\sqrt{x}}{\sqrt{x}-2}\)
4. Rút gọn D= \(\dfrac{\sqrt{x}-1}{\sqrt{x}-2}\)-\(\dfrac{5\sqrt{x}-8}{x-2\sqrt{x}}\)
1: \(A=\dfrac{\sqrt{x}+1}{\sqrt{x}-1}+\dfrac{\sqrt{x}-1}{\sqrt{x}+1}-\dfrac{3\sqrt{x}+1}{x-1}\)
\(=\dfrac{\left(\sqrt{x}+1\right)^2+\left(\sqrt{x}-1\right)^2-3\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x+2\sqrt{x}+1+x-2\sqrt{x}+1-3\sqrt[]{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{2x-3\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{\left(2\sqrt[]{x}-1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{2\sqrt{x}-1}{\sqrt{x}+1}\)
2: \(B=\dfrac{\sqrt{x}}{\sqrt{x}+2}-\dfrac{2\sqrt{x}-4}{x-4}\)
\(=\dfrac{\sqrt{x}}{\sqrt{x}+2}-\dfrac{2\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{\sqrt{x}}{\sqrt{x}+2}-\dfrac{2}{\sqrt{x}+2}=\dfrac{\sqrt{x}-2}{\sqrt{x}+2}\)
3: \(C=\dfrac{x}{x-4}+\dfrac{1}{\sqrt{x}-2}+\dfrac{1}{\sqrt{x}+2}\)
\(=\dfrac{x+\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{x+2\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{\sqrt{x}}{\sqrt{x}-2}\)
4: \(D=\dfrac{\sqrt{x}-1}{\sqrt{x}-2}-\dfrac{5\sqrt{x}-8}{x-2\sqrt{x}}\)
\(=\dfrac{\sqrt{x}-1}{\sqrt{x}-2}-\dfrac{5\sqrt{x}-8}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)-5\sqrt{x}+8}{\left(\sqrt{x}-2\right)\cdot\sqrt{x}}=\dfrac{x-6\sqrt{x}+8}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-4\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}-4}{\sqrt{x}}\)
1) \(...\Rightarrow A=\dfrac{\left(\sqrt{x}+1\right)^2+\left(\sqrt{x}-1\right)^2-3\sqrt{x}-1}{x-1}\left(x\ne1\right)\)
\(\Rightarrow A=\dfrac{x+2\sqrt{x}+1+x-2\sqrt{x}+1-3\sqrt{x}-1}{x-1}=\dfrac{2x-3\sqrt{x}+3}{x-1}\)
\(\Rightarrow A=\dfrac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}=\dfrac{2\sqrt{x}-1}{\sqrt{x}+1}\)
2) \(...\left(x\ne4\right)\Rightarrow B=\dfrac{\sqrt{x}\left(\sqrt{x}-2\right)-2\sqrt{x}+4}{x-4}=\dfrac{x-4\sqrt{x}+4}{x-4}=\dfrac{\left(\sqrt{x}+2\right)^2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}+2}{\sqrt{x}-2}\Rightarrow\left(đpcm\right)\)
3) \(...\left(x\ne4\right)\Rightarrow C=\dfrac{x+\sqrt{x}+2+\sqrt{x}-2}{x-4}=\dfrac{x+2\sqrt{x}}{x-4}=\dfrac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}}{\sqrt{x}-2}\left(đpcm\right)\)
4) \(...\left(x\ne0;4\right)\Rightarrow D=\dfrac{\sqrt{x}-1}{\sqrt{x}-2}-\dfrac{5\sqrt{x}-8}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{x-\sqrt{x}-5\sqrt{x}+8}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{x-6\sqrt{x}+9-1}{\sqrt{x}\left(\sqrt{x}-2\right)}\)
\(\Rightarrow D=\dfrac{\left(\sqrt{x}-3\right)^2-1}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-4\right)}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}-4}{\sqrt{x}}\)
